Chen primes in arithmetic progressions Pawe l Karasek Abstract 6 Wefinda lower bound for thenumberof Chen primes in thearithmetic progression amodq, 1 where (a,q)=(a+2,q)=1. Ourestimate is uniform for q≤logMx,where M >0 is fixed. 0 2 n a 1 Introduction J 1 3 The famoustwinprimeconjectureassertsthatthereexistinfinitely manyprimespsuchthatp+2 ] is also a prime. It is a common perception that our current methods are insufficient to prove this T supposition, although in some way one is able to get very close to it. For example in [1] and [2], J. N Chen proved the following famous result. . h t Theorem 1.1. There are infinitely many primes p such that p+2 has at most two prime factors, a each greater or equal than p1/10. m [ Infact,Chenshowedthatthenumberofsuchprimesinaninterval[1,x]isgreaterthancx/log2x, where c is some positive constant. In [3], B. Green and T. Tao showed that there are infinitely many 2 v 3-term arithmetic progressions among them. This result was extended by B. Zhou in [4], where it is 3 proven that in the same circumstances one can find arithmetic progressions of any given length. 7 ItisnaturaltoaskwhetherChenprimes(ortwinprimes)areequallydistributedamongarithmetic 8 progressions as primes do. For example, using Cramér’s random model we are led to the following 2 conjecture about twin primes (cf. section 2 for the notation). 0 . 1 Conjecture 1.2. Let a,q be positive integers such that (a(a+2),q)=1. Then 0 6 x #{twin primes p : p≡a (mod q)}=(2Π +o (1)) . 1 2 q ϕ (q)log2x 2 : v i The goal of this paper is to prove a uniform (in a Siegel−Walfisz manner) lower bound for an X analogous count of Chen primes. We will accomplish this using techniques developed by Chen with r slight modifications (specifically, we shall follow the discussion in [7]). The main result can be stated a as follows: Theorem 1.3. Let M > 0 and let a,q be positive integers such that (a,q) = (a +2,q) = 1 and q ≤logMx. Then x Λ (n)1 (n+2)1 ≫ . a,q P2 (n+2,P(x1/8))=1 M ϕ2(q)logx x/2≤Xn≤x−2 After removing the small contribution of the prime powers, we can also convert this into a more elegant form. 1 1 INTRODUCTION Corollary 1.4. Let M > 0 and let a,q be positive integers such that (a,q) = (a+2,q) = 1 and q ≤logMx. Then x 1 ≫ . n∈PCh M ϕ (q)log2x n6x 2 X n a (q) ≡ The proof of Theorem 1.3 is based on the following observation. Lemma 1.5. For x2/3 p1/8, p+2=p p } ∪ {twin primes}, Ch p1,p2∈P 1 2 1 2 • Π := (1− 1 )≈0.66, 2 p>2 (p 1)2 − Q • ϕ (n):=n 1− 2 ×(1−1 /2), 2 pn:p=2 p 2n | 6 | (cid:16) (cid:17) • ∆(f;a (d))Q:= f(n)− 1 f(n), n a(d) φ(d) n:(n,d)=1 ≡ • we will treat MPas an absolute conPstant, so in the further sections the dependence on M will not be emphasized in any way. 3 Preliminaries Theorem 3.1 (Siegel−Walfisz). For any A>0 there exists a constant C >0 such that A x Λ(n)= +O(xexp(−C logx)) A φ(d) n x n=Xa≤(d) p for every residue class a (d) satisfying (a,d)=1 and for all x≥2 such that d≤logAx. Theorem 3.2 (Bombieri−Vinogradov). Let x≥2. Then max ∆(Λ1[1,x];a (d)) ≪A xlog−Ax d Da∈(Z/dZ)× X≤ (cid:12) (cid:12) (cid:12) (cid:12) for all A>0, where D ≤x1/2log−Bx for some sufficiently big B =B(A). Let us define the real functions f(s) and F(s) in the following way: 2eγ 2eγ F(s)= , f(s)= log(s−1). s s Theorem 3.3 (Jurkat−Richert). Consider s > 1 and z,D ≥ 2 which satisfy z = D1/s. Let Q be a finite subset of P and let Q be the product of the primes in Q. Furthermore, let h be a multiplicative function that for some ε with 0<ε<1/200 satisfies the inequality logz (1−h(p)) 1 <(1+ε) (3.1) − logu pu∈YPp(cid:31)0 and some error terms r . Then, for any 10 provided that D ≤x1/2log−Bx f(cid:12)(cid:12)or some sufficiently large B =(cid:12)(cid:12)B(A). Proof. Follows from Theorem 22.3 from [5]. We also note that for each integer d ≥ x1/8 we have b(n)≪x7/8,sothe totalcontributionofingredientsofthisformisatmostx7/8+o(1) which n:(n,d)>1 is neglible. P Lemma 3.6. Let d ≥ 1 be a fixed integer and let f : (0,∞)d → C be a fixed compactly supported, Riemann integrable function. Then for x>1 we have 1 logp logp dt ...dt 1 d 1 d f ,..., = f(t ,...,t ) +o (1). 1 d x p1X,...,pd p1...pd (cid:18)logx logx (cid:19) Z(0,∞)d t1...td →∞ Proof. Follows from Mertens’ second theorem combined with elementary properties of the Riemann integral. 4 4 ESTIMATING A 1 4 Estimating A 1 According to Theorem 3.3 let us define E as the set of positive integers from the residue class d −2 (d). Let us also define a multiplicative function g(d) such that g(2) = 0 and g(p) = 1/(p−1). The precise value in non-square-free numbers is not relevant. Put z = x1/8 and D = x1/2−ǫx, where ǫ :=(logx) 1/2. WetakeQtobethelargestsetoftheformasinRemark3.4satisfyingQ≤loglogx. x − We have A = Λ (n). 1 a,q x/2≤Xn≤x−2 n6∈Sp

Chen primes in arithmetic progressions

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